I have used K240Ms for several years now as my main headphone, and recently switched to K701s. Over the years with the K240Ms, they were unamped, using the headphone out of the Behringer mixer. I purchased a Corda Aria that arrived roughly a week before my K701s and used this amp to drive the 240M for that week.
First, the K240M is hard to drive, sure. I get more volume from my K701s unamped than I did from my K240M by a long shot. At the same time it does seem that amping adds a bit more to the K701 than it did to the K240. I am not sure how to explain it, the 240M have much more volume when amped, and do open up a bit more, but do not seem to have a ton more punch when compared to the K701s.
The K701s offer plenty of volume when unamped, but adding the Aria just seems to make them open up. This may well be the difference, as I could get enough volume unamped, I was not listening to them much louder amped. While the volume was similar everything just seemed much more controlled, much cleaner. It may just be that the Aria is not enough of an amp for the 240M (which says a lot considering people say the K701 is power hungry) and something with a bit more power would open them up as well.
As for the final comparisons between the sound of the two headphones, the K240M is a solid headphone for the price, the mids were great, everything was fairly clear, but the K701 has much more extension on both ends and just more detail throughout. I still use my 240M for assorted computer audio, but all of my critical listening is on my K701s